Method of generating complex waveforms and modulating signals

ABSTRACT

Methods for generating complex waveforms, including step functions, impulse functions, and gate pulses are provided, as well as methods for generating modulated waveforms employing a number of known and newly developed modulation formats. The methods of the present invention employ a continuous linear function, wherein all output points are defined. Discontinuities and singularities are eliminated, yet pulses having nearly instantaneous transitions may be achieved. Thus, gate pulses step functions, binary waveforms and the like may all be generated from a single function, where they entire output range of the function is defined over a continuous input domain.

BACKGROUND OF THE INVENTION

[0001] The present invention relates to methods for generating complexwaveforms and modulated signals. Using conventional mathematicaltechniques, it is difficult to express many output functions commonlyused in telecommunications, computing, machine control and otherelectronic systems. For example, using conventional techniques it isimpossible to express a single square pulse having a value of one over ashort input range and value of zero elsewhere. At least two equationsare required (y=1, y=0) along with restrictions as to the locationswhere one equation or the other holds sway.

[0002] Another problem in mathematically expressing square pulses,binary bit streams and other common waveforms is the mathematicaldifficulty in expressing discontinuities or singularities such as thosethat occur at the transitions between 0 and 1 and from 1 to 0 in binarysignals. Theoretically these transitions take place instantaneously, theoutput function being undefined at the point of transition. Practically,this can lead to uncertainty when such functions are applied toelectronic circuits and the like. Intermediate values can be recorded atthe transitions, leading to ambiguous or inaccurate results.

[0003] Such singularities occur in many commonly used waveforms. Stepfunctions, gate pulses, impulse functions, and the like, all raisedifficult problems of mathematical expression for scientists andengineers who work with such waveforms everyday. In fact, solutions tothese types of problems must be designed into the hardware circuitry ofmany digital electronics devices. Timing circuits, error correctingcircuits and the like are necessary to verify data and synchronizesignals to ensure that bits are not being read during binarytransitions, for example. This adds to the cost, complexity and size ofmany devices.

[0004] Conventional mathematics for encoding data onto binary and othersignals are also complex. Many modulation techniques such as pulse widthmodulation, time division multiplexing (TDM), code division multipleaccess, and the like, all require extensive mathematical algorithms toimplement. Again, the complexity of the mathematics results in complexhardware and software solutions for generating and using such signals.Synchronization is also a problem with many current digital systems.Ensuring that bits are read at the proper times and other timing andphase shift issues can greatly add to the complexity of digital systems.

[0005] Finally, because the binary transitions in digital signals andsingularities in other discontinuous functions are undefined, theyrepresent a portion of a signal that may not be used for carryinginformation. The density of data carried by a binary signal can begreatly increased if it is possible to mathematically define thetransitions between ones to zeroes and zeroes to ones in an easyconvenient manner and in such a way that the transitions themselves arecapable of carrying data.

[0006] In light of the above, there presently exists a need for improvedmethods of generating and expressing complex waveforms. Improved methodsof generating waveforms should accommodate waveforms which havetraditionally been difficult to express mathematically such as stepfunctions, gate functions and the like. Furthermore, the output positionof such functions should be controlled with great precision so that itmay be accurately predicted when pulses should or should not occur.

[0007] New modulation techniques are also needed for implementing suchimproved methods of generating complex waveforms. Ideally, suchtechniques should allow every aspect of a complex waveform to bedetermined by either a single function or by cascading a series of likefunctions. Furthermore, such improved techniques should allow formultiple modulation schemes to be added to a single carrier so that thedensity of data stored on the carrier may be greatly increased.

SUMMARY OF THE INVENTION

[0008] The present invention relates to methods of generating complexwaveforms and modulated output signals. A linear and continuous functionis provided by which any number of output waveforms may be expressedusing a single equation. Using the present technique even waveformswhich have traditionally resisted convenient mathematical expression dueto discontinuities and singularities may be expressed and manipulated inan elegant straight forward manner.

[0009] According to the present invention complex continuous waveformsmay be generated using a single function wherein an output value of thewaveform is unambiguously defined for every input value of a continuousinput domain. According to the invention, the output waveform includesat least one operational window. The significance of the operationalwindow is that the output waveform is substantially equal to zero orsome other base value at input points outside the operational window.For input points within the operational window the output waveform isequal to the value of a specified window function. An important featureof the present invention is that the position of one or more suchoperational windows may be controlled with great precision. A positionlocus function establishes a base position for each operational window,and a phase shift parameter determines the actual position of theoperational window relative to this base position or locus.

[0010] The method includes the step of providing a continuous linearfunction such as the DNAX function disclosed herein. The functionprovided includes a number of input parameters that can be manipulatedin order to provide a desired output waveform. In an embodiment of theinvention the parameters include a phase shift parameter d for adjustingthe phase of the one or more operational windows relative to the zerocrossings of a position locus function. A second parameter n(x)determines the slew rate, the speed of the transitions between outputvalues of the generated waveform corresponding to input values outsidethe operational window and input values within the operational window. Athird parameter a(x) determines the size of the operational windowaperture, and a fourth parameter X(x) establishes the position locus ofthe operational window. A selectable window function ƒ(x) determines theshape of the output waveform within the operational window and aweighting factor b(x) may be provide which affects the linearity of thewindow function ƒ(x) within the operational window.

[0011] The second step in the process is selecting appropriate valuesfor the above described parameters and functions to generate an outputwaveform of the desired shape using the single function. Such waveformsmay be added subtracted, multiplied, divided or input to additionalfunctions including additional DNAX functions, adding further complexityto the shape of the desired output waveform.

[0012] The present invention may also be employed to generate one ormore pulses. Again, a continuous function is provided. The functionincludes a position locus parameter for determining the position of eachof the one or more pulses, an aperture parameter for determining thewidth of each pulse, and a slew rate parameter for determining the rateat which the transitions at the edges of the one or more pulses occur.The amplitude of each of the one or more pulses can be determined by awindow function or may be set equal to a constant. The position locusparameter may then be set equal to some function having at least onex-intercept, such that pulses occur at positions related to the one ormore x-intercepts of the position locus function. By applyingappropriate values to the input parameters gate pulses, step pulses,impulses and other pulse shapes may be generated. Furthermore, applyinga function having multiple x-intercepts such as a periodic sinewave orsome other non-periodic function having multiple x-intercepts as theposition locus function causes a whole series of pulses to be formed.Each pulse is located based on the position of the zero crossings of theposition locus function.

[0013] The present invention employing the DNAX function may also beemployed to generate waveforms that have been modulated according to anumber of different modulation techniques. For example, pulse widthmodulation, pulse position modulation, pulse amplitude modulation, aswell as time division multiplexing, code division multiple access, andother known signaling techniques may be implemented with an ease whichheretofore has not been achieved. Furthermore, new modulation techniquessuch as slew rate modulation and high volume high density modulation arealso made possible by the inventive methods described and claimed in thepresent specification.

[0014] Additional features and advantages of the present invention aredescribed in, and will be apparent from, the following DetailedDescription of the Invention and the figures.

BRIEF DESCRIPTION OF THE FIGURES

[0015]FIG. 1 is a plot showing an example of a complex output waveformhaving the values ${y(x)} = \left\{ \begin{matrix}{{0\quad {for}\quad {x}} > 4} \\{{x^{2}{\quad \quad}{for}\quad {x}} \leq 4}\end{matrix} \right.$

[0016]FIG. 2 is a functional block representation of a DNAX function orcell.

[0017]FIG. 3 is a plot showing an output waveform generated using theDNAX function having an operational window 1 unit wide, centered at 0,and having a sinusoidal output within the operational window.

[0018]FIG. 4 is a plot showing a waveform generated using the DNAXfunction similar to the waveform shown in FIG. 3, wherein the frequencyof the sinusuidal waveform within the operational window has beentripled, and the amplitude has been multiplied by 8.

[0019]FIG. 5 is a plot showing a rectangular pulse generated using theDNAX function having a pulse width of 2.

[0020]FIG. 6 is a plot showing another rectangular pulse generated usingthe DNAX function, but with a pulse width of 1.

[0021]FIG. 7 is a plot showing another rectangular pulse generated usingthe DNAX function, but with a pulse width of {fraction (1/8)}.

[0022]FIG. 8 is a plot showing a rectangular pulse train generated usingthe DNAX function, the plot also includes the position locus functionπ/2 Sin2πx input to the position locus parameter X(x) of the DNAXfunction.

[0023]FIG. 9 is a plot showing a rectangular pulse {fraction (1/8)} widephase shifted 0.5.

[0024]FIG. 10 is a plot showing a series of rectangular pulses ⅛ widephase shifted 0.7.

[0025]FIG. 11 is a plot showing several superimposed rectangular pulsesgenerated using the DNAX function, each pulse was created with adifferent value of the slew rate parameter n(x).

[0026]FIG. 12 is a plot of a waveform generated using the DNAX functionwith no weighting factor (b(x)=1).

[0027]FIG. 13 is a plot showing a waveform generated using the DNAXfunction similar to the waveform shown in FIG. 10, but with a positiveweighting factor greater than one, (b=0.5).

[0028]FIG. 14 is a plot showing a waveform generated using the DNAXfunction similar to the waveform shown in FIG. 10, but with a positiveweighting factor less than one, (B=2).

[0029]FIG. 15 is a plot showing a pulse width modulated output waveformgenerated using the DNAX function.

[0030]FIG. 16 is a plot showing the modulating signal for generating thePWM waveform of FIG. 15.

[0031]FIG. 17 is a plot of the unmodulated pulse train generated usingthe DNAX function prior to being modulated by the modulating signalshown in FIG. 14, along with the position locus functionX(t)=Sin(2π·50x10³t).

[0032]FIG. 18 is a graphical representation of the linear transferfunction for determining the width of pulses in the PWM waveform of FIG.15 in relation to the corresponding magnitude of the modulating signals(t) shown in FIG. 16.

[0033]FIG. 19 is a plot showing the position locus functionX(t)={fraction (1/3)}Sinωt and the absolute value of its derivative${{\frac{\quad}{x}\frac{1}{3}{{Sin}\omega}\quad t}}.$

[0034]FIG. 20 is a plot showing the modulating index function which isinput to the operational window aperture function of the DNAX functionto modulate the width of the output pulses according to the modulatingsignal s(t) which is also shown for reference.

[0035]FIG. 21 is the modulation index function of FIG. 20 with aconstant c=π/4 added thereto.

[0036]FIG. 22 is a plot showing a rectangular modulating signalgenerated using the DNAX function to be input as the slew rate parametern(t) in a second DNAX function to implement slew rate modulation (SRM)along with the position locus function X(t) for generating therectangular pulses.

[0037]FIG. 23 is a plot of a slew rate modulated waveform generatedusing the DNAX function, with the waveform shown in FIG. 22 input as theslew rate parameter n(t).

[0038]FIG. 24 is a plot of a slew rate modulated waveform generatedusing the DNAX function in substantially the same manner as the waveformshown in FIG. 23, but wherein the phase of the rectangular input to theparameter n(t) has been adjusted such that both the leading and trailingtransitions of each pulse fall within areas corresponding to relativelyhigh slew rate.

[0039]FIG. 25 is a plot of a slew rate modulated waveform generatedusing the DNAX function in substantially the same manner as thewaveforms shown in FIGS. 23 and 24, but wherein the phase of therectangular input to the slew rate parameter has been adjusted such thatthe leading edge of each pulse is located in an area corresponding to anarea of relatively low slew rate and the trailing edges are located inareas corresponding to relatively high slew rate.

[0040]FIG. 26 is a plot of a slew rate modulated waveform using the DNAXfunction in substantially the same manner as the waveforms shown inFIGS. 23, 24 and 25, but wherein the operational window apertureparameter a(t) of the first DNAX function has been adjusted to reducethe duty cycle of the modulating signal pulses n(t), such that both theleading and trailing edges of each pulse of the output waveform y(t)falls in an area corresponding to relatively low slew rate.

[0041]FIG. 27 shows a plurality of different pulse shapes that may beachieved using slew rate modulation according to the present invention.

[0042]FIG. 28 shows a functional block diagram for performing slew ratemodulation using the DNAX function according to the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0043] The present invention relates to methods of generating complexwaveforms and signal modulation techniques for use in communications,signal processing, manufacturing, and other applications. The methods ofthe present invention are based on the application of a unique functionwhich has been given the name DNAX. DNAX is an acronym formed by theletters designating four separate input parameters of the DNAX functionwhich govern the shape of the output waveform. By astute manipulation ofthese input parameters and selectively rearranging the form of the DNAXfunction, nearly any desired output waveform can be mathematicallygenerated via a single mathematical expression.

[0044] The DNAX function is a continuous linear function that provides adefined output value for every input value of a continuous input domain.Using the DNAX function, complex waveforms including waveforms havingsharp discontinuities which have heretofore defied easy mathematicalexpression, such as step functions, impulse functions, gate pulses, andbinary pulse sequences, may be expressed using a single equation.Furthermore, the mathematical difficulties of dealing with undefinedsingularities such as the step function or impulse function aresubstantially eliminated.

[0045] The DNAX function is based on the concept of an operationalwindow. FIG. 1 shows a Cartesian coordinate system 10 on which DNAXoutput waveform 12 is plotted. The output waveform 12 includes threedistinct regions. The first region 14 extends from −∞ to −4. The secondregion 16 extends from −4 to +4. And the third region 18 extends from +4to +∞. Within the first and third regions 14, 18 the value of the outputwaveform 12 is substantially equal to 0. In the second region 16,however, the value of the output waveform 12 is equal the square of theinput value, or y(x)=x². As will be described below, the output waveform12 may be expressed with a single DNAX function. The operational windowis the second region 16 where the output waveform takes on a non-zerovalue. The DNAX function allows the creation of output waveforms havingany number of operational windows. What is more, the DNAX functionallows for precise control over the position and width of suchoperational windows and the shape of the output waveform withinoperational windows.

[0046] Using conventional mathematical techniques at least two equationsare required to describe the waveform 12 shown in FIG. 1:${y(x)} = \left\{ \begin{matrix}{{0\quad {for}\quad {x}} > 4} \\{{x^{2}{\quad \quad}{for}\quad {x}} \leq 4}\end{matrix} \right.$

[0047] Using this mathematical description, the transitions at +4,between 0 (the value of the output waveform in the first and thirdregions 14, 18) and 16 (the value of the function y=x² at the end pointsof the operational window are undefined. The vertical line segments atthe boundaries between the first and second and second and third regionsrepresent an output function of infinite slope. It is impossible todefine the value of the output function y(x) at these transitions usingconventional techniques without adding additional complexity to theequations that define the output waveform 12. Using the DNAX function,however, the output waveform illustrated in FIG. 1 can be described in amanner such that all output points are uniquely defined with but asingle mathematical function.

[0048]FIG. 2 shows the DNAX function in block form. Four inputparameters d, n, a and x are input into the DNAX function 24. The DNAXfunction 24 outputs a continuous waveform DNAX (x) having uniquelydefined output values for all input values x. The shape of the outputwaveform is determined by on the values supplied to the input parametersd, n, a, and x, as well as a window function ƒ(x), and a weightingfactor b(x) if desired. The input parameters n, a, and x are moreappropriately defined as n(x), a(x), and x(x) because, according tovarious embodiments of the invention, variable functions may be input tothese parameters rather than constants.

[0049] The general DNAX function may be expressed in two differentforms. These are: $\begin{matrix}{{{y(x)} = \left\lbrack {\left( {{a(x)}{X\left( {x - d} \right)}} \right)^{n{(x)}} + {f(x)}} \right\rbrack^{\lbrack{{b{(x)}} - {({{a{(x)}}{X{({x - d})}}})}^{n{(x)}}}\rbrack}}{and}} & {1A} \\{{y(x)} = {{f(x)}\left\lbrack \left\lbrack {\left( {{a(x)}{X\left( {x - d} \right)}} \right)^{n{(x)}} + 1} \right\rbrack^{\lbrack{{b{(x)}} - {({{a{(x)}}{X{({x - d})}}})}^{n{(x)}}}\rbrack} \right\rbrack}} & {1B}\end{matrix}$

[0050] The parameter X(x−d) is the position locus parameter, whichdefines the location of the one or more operational windows along theabscissa axis. The DNAX function generates an operational windowwhenever X(x−d)=0. The parameter d establishes a phase shift which, whend is not equal to zero, causes the position of the operational window toshift away from the x intercepts of the position locus function X(x) byan amount equal to the value of d. When d equals 0, there is no phaseshift and the DNAX function creates an operational window or windowscentered on the abscissa intercepts, or zero crossings of the positionlocus function X(x). Otherwise, the operational window (or windows asthe case may be) are shifted in phase by amount equal to the value of d.

[0051] The parameter a(x) defines the width of the operational window.Pulse duration or pulse width is defined according to the DNAX techniqueby the following equation:${{Pd}(x)} = {\frac{2}{{a(x)} \cdot {{\frac{\quad}{x}{X\left( {x - d} \right)}}}}.}$

[0052] Thus, pulse duration is inversely proportional to the value ofthe operational window aperture parameter a(x) and the absolute value ofthe derivative of the position locus function X(x−d) at the positionwhere X(x−d) crosses the x axis.

[0053] n(x) represents the slew rate parameter. This parameterestablishes the slope of the transitions into and out of the operationalwindow. As the value of n(x) increases, the slope of the transitionstends toward infinity, providing faster and faster transitions.Nonetheless, even with extremely large values of n(x), resulting in nearinstantaneous transitions into and out of the operational window, everypoint on the output waveform is uniquely defined. Depending on theapplication, n(x) may be a variable function or a constant. Preferably,in order to achieve well behaved output waveforms, n(x) is limited toeven constants in those applications where a modulating function is notapplied to the n(x) parameter.

[0054] The function ƒ(x) represents the window function which definesthe output waveform generated by the DNAX function for input values thatare within the operational window. The output value of the DNAX functionwill be substantially equal to 0 or some other base value forsubstantially all input values outside the operational window. On, theother hand, the output value of the DNAX function will be substantiallyequal to the output value of the function ƒ(x) for substantially allinput values of x within the range of the operational window. Thetransitions from the base value (typically 0) to ƒ(x) at the leading andtrailing edges of the operational window are governed by the slew rateparameter n(x).

[0055] The factor b(x) in the exponent term in the equations 1A and 1Bis a weighting factor. This factor adds a predefined amount ofdistortion to the output waveform ƒ(x) within the operational window.Weighting factors and pre-planned distortion will be described in moredetail below. b(x)=1 corresponds to no weighting and no distortion addedto the output function ƒ(x). For the remainder of this discussion,unless explicitly stated otherwise, the weighting factor b(x) will beset equal to 1. Thus, there will be no distortion added to the outputwaveform due to intentional weighting.

[0056] Having introduced the DNAX function, we will apply the generalform of the DNAX function to generate the output waveform 12 shown inFIG. 1. The DNAX input parameters are set as follows: $\begin{matrix}{d = 0} \\{{n(x)} = 100} \\{{a(x)} = \frac{1}{4}} \\{{X(x)} = x} \\{{f(x)} = x^{2}}\end{matrix}$

[0057] Thus, the general form of the DNAX function becomes${y(x)} = \left\lbrack {\left( \frac{x}{4} \right)^{100} + x^{2}} \right\rbrack^{\lbrack{1 - {(\frac{x}{4})}^{100}}\rbrack}$

[0058] The general shape of the output waveform may be verified byplotting a small sample of points as shown in the table below. x x² DNAx(x²) −10.00 100.00 0.000000000000000 −5.00 25.00 0.000000000000000 −4.1016.81 0.000000000000000 −4.05 16.40 0.000634211875248 −4.01 16.080.445047557071670 −4.0000000000000000 16.00000000000000001.0000000000000000 −3.9998575555555500 15.99886046473480001.0101215209168800 −3.9998575555000000 15.99886046429040001.0101215248765600 −3.8999999999999900 15.209999999999900012.3087511446208000 −3.00 9.00 8.999999999993980 −2.50 6.256.250000000000000 −2.00 4.00 4.000000000000000 −1.50 2.252.250000000000000 −1.00 1.00 1.000000000000000 −0.50 0.250.250000000000000 0.00 0.00 0.000000000000000 0.50 0.250.250000000000000 1.00 1.00 1.000000000000000 1.50 2.252.250000000000000 2.00 4.00 4.000000000000000 2.50 6.256.250000000000000 3.00 9.00 8.999999999993980 3.899999999999990015.2099999999999000 12.308751144620800 3.999857555500000015.9988604642904000 1.010121524876560 3.999857555555550015.9988604647348000 1.010121520916880 4.000000000000000016.0000000000000000 1.000000000000000 4.01 16.08 0.445047557071670 4.0516.40 0.000634211875248 4.10 16.81 0.000000000000000 5.00 25.000.000000000000000 10.00 100.00 0.000000000000000

[0059] From this small sample it is evident that outside the operationalwindow, i.e. for values of x less than −4.00 and greater than +4.00 theoutput value of the DNAX function is substantially 0. For values of xbetween −4.00 and +4.00 the output value of the DNAX function issubstantially equal to x². Furthermore, at the edges of the operationalwindow (±4) the transitions 20, 22 between 0 outside the operationalwindow and with the operational window x² occur very rapidly. In fact,even when the input value of x is within 0.05 of the transition ateither edge of the operational window, the output value of the DNAXfunction is still substantially equal to zero. At the exact edge of theoperational window, DNAX equals 1 and within the operational window theoutput values of the DNAX function rapidly approach x². Nonetheless theoutput of the DNAX function is defined throughout the transitions. Itshould be noted that even these seemingly insignificant deviations fromthe desired output values at the transitions can be reduced stillfurther (or made larger if so desired) by simply increasing ordecreasing the value of the slew rate parameter n(x).

[0060]FIGS. 3 and 4 illustrate DNAX output waveforms wherein the windowfunction ƒ(x) has been changed to a sine wave. The two waveforms areidentical except for the frequency and amplitude of the sine wave thatappears within the operational window. The parameters entered into thegeneral form of the DNAX function to achieve the waveforms shown inFIGS. 3 and 4 were as follows:

[0061] Thus, both waveforms 26, 34 of FIG. 3 and 4 have a singleoperational window 30, 38 respectively, centered around the originextending from −0.5 to +0.5. Furthermore, each waveform 26, 34 includesfirst regions 28, 36 and second regions 32, 40 defined by input valuesoutside the operational windows where the DNAX functions' output valuesare substantially zero. Within the operational window 30 of the outputwaveform 26 of FIG. 3 the output of the DNAX function equals Sin(20πx).Within the operational window 38 of the output waveform 34 of FIG. 4 theoutput of the DNAX function equals 8 Sin(60πx). Thus, within theoperational windows the output waveforms are sine waves havingfrequencies of 10 Hz and 30 Hz respectively. Thus, the frequency of thewaveform displayed within the operational window 38 of waveform 34 istriple that of the waveform displayed in the operational window ofwaveform 26. What is more, the amplitude of waveform 34 has beenincreased by a factor of 8. In fact, an important feature of the DNAXfunction is that there are not limits on the magnitude of the outputwaveform within the operational window. Any amplitude is possible. Itshould be noted that the changes brought about between the outputwaveform of FIG. 3 and that of FIG. 4 were accomplished by simplychanging the window function ƒ(x), a single input parameter of thegeneral DNAX function.

[0062] Next, we consider the effects of varying the value of theoperational window aperture parameter a(x). In employing the DNAXfunction, we note that pulse duration is defined as${{Pd}(x)} = {\frac{2}{{a(x)} \cdot {{\frac{}{x}{X\left( {x - d} \right)}}}}.}$

[0063] Thus, pulse duration is inversely proportional to the value ofthe parameter a(x) and the absolute value of the derivative of theposition locus function X(x−d). Again, we start with the general form ofthe DNAX function:y(x) = [a(x) ⋅ X(x)^(n(x)) + f(x)]^([1 − (a(x)X(x))^(n(x))])

[0064] we set the DNAX input parameters as follows: $\begin{matrix}{d = 0} \\{{n(x)} = 10^{3}} \\{{a(x)} = 1} \\{{X(x)} = x}\end{matrix}$

[0065] Further, we set the window function ƒ(x) equal to a constant, inthis case ƒ(x)=1, so that the output of the DNAX function will be 1within the operational window and 0 outside the operational window. Withthese values the DNAX function reduces toy(x) = [(x)^(10³) + 1]^([1 − (x)^(10³)])

[0066] resulting in the output waveform 42 shown in FIG. 5.

[0067] As can be seen, this combination of parameters results in aoutput waveform which includes a single rectangular pulse of amplitude 1centered on the origin and extending from −1.0 to +1.0. As with theprevious examples, the transitions from 0 to 1 and from 1 to 0 are verysteep due to the high slew rate n(x)=1000. Nonetheless, the transitionsare continuous and all points on the output waveform are uniquelydefined.

[0068] The duration of the pulse is 2 due to the relationship betweenpulse duration and the operational window aperture parameter a(x) andthe absolute value of the derivative of the position locus functionX(x). Since at x=0, a(x)=1, X(x)=0, and ${{\frac{{X(x)}}{x}} = 1},$

[0069] the equation for the pulse duration at x=0 becomes${Pd} = {\frac{2}{{a(x)}{\frac{{X\left( {x - {xd}} \right)}}{x}}} = {\frac{2}{1 \cdot 1} = 2.}}$

[0070] Thus, the operational window is centered around zero and has endpoints at −1 and +1 resulting in the pulse duration of 2.

[0071]FIG. 6 shows another DNAX output waveform 44. Again a rectangularpulse of amplitude 1 centered on the origin is generated. The width ofthe pulse in FIG. 6, however, is half the width of the pulse in FIG. 5.The waveform 44 of FIG. 6 was obtained with the same DNAX functionparameters except that the operational window aperture parameter a(x)was changed from a(x)=1 to a(x)=2. Thus,${{Pd} = {\frac{2}{{a(x)}{\frac{{X\left( {x - {xd}} \right)}}{x}}} = {\frac{2}{2 \cdot 1} = 1}}},$

[0072] and the pulse contained in output waveform 44 begins at −0.5 andends at +0.5.

[0073]FIG. 7 shows yet another output waveform, 46. In this example, theoperational window aperture parameter has been set to a(x)=16. Thisleads to the narrow $\frac{1}{8}$

[0074] duration pulse having transitions at −0.0625 and +0.0625.

[0075] Next, we will consider the DNAX function employed to generate acontinuous pulse sequence such as the periodic rectangular pulsewaveform 48 shown in FIG. 8. In this example, the DNAX parameters havebeen set as follows: $\begin{matrix}{{d(x)} = 0} \\{{n(x)} = 10000} \\{{a(x)} = 16} \\{{X(x)} = \frac{{Sin}\left( {2\pi \quad x} \right)}{2\pi}} \\{{and}\quad} \\{{f(x)} = 1.}\end{matrix}$

[0076] Thus, the general form of the DNAX function becomes${y(x)} = \left\lbrack {\left( {\frac{8}{\pi}{{Sin}\left( {2\pi \quad x} \right)}} \right)^{10000} + 1} \right\rbrack^{\lbrack{1 - {({\frac{8}{\pi}{{Sin}{({2\pi \quad x})}}})}^{10000}}\rbrack}$

[0077] Note that the input parameters just described are the same asthose for producing the waveform 46 having the narrow ⅛ wide pulse inFIG. 7, except that the value of the position locus parameter X(x) hasbeen replaced with a periodic function. In order to produce the singlepulse waveform 46 of FIG. 7, the position locus parameter X(x) was setto X(x)=x (a line having a single x intercept at the origin and a slopeequal to one.) whereas to produce the repeating pulses of outputwaveform 48 of FIG. 8, the position locus parameter is set equal to${(x) = \frac{{Sin}\left( {2\pi \quad x} \right)}{2\pi}},$

[0078] a periodic function for which the absolute value of itsderivative is equal to 1 when the function itself is equal to 0. Thisposition locus function is plotted in FIG. 8. The sinusoidal positionlocus waveform 50 includes a plurality of regularly spaced zerocrossings at 0, 0.5, 1.0, 1.5 . . . etc. (P_(i)(x_(i):0)).

[0079] An important aspect of the DNAX function is that it generates anoperational window (and in this case a pulse) corresponding to eachposition where the position locus parameter X(x) equals zero. If nophase shift is applied (d=0) the operational window (pulse) will becentered around the zero crossings. Therefore, with${{X(x)} = \frac{{Sin}\left( {2\pi \quad x} \right)}{2\pi}},$

[0080] pulses are generated at 0, 0.5, 1, 1.5, 2.0, etc., and the outputwaveform 48 is created. In this case, the pulses are substantially thesame as the single pulse shown in FIG. 7, with an amplitude of 1(ƒ(x)=1) and since${{\frac{\quad}{x}\left( \frac{\sin \quad 2\quad \pi \quad x}{2\pi} \right)}} = {1\quad {at}\quad {P_{i}\left( {x_{i}:0} \right)}}$

[0081] and a(x)=16, the pulses have a width of ⅛ with steep relativelyfast transitions due to the relatively high value of the slew rateparameter n(x)=10000.

[0082] Returning to the single narrow pulse shown in FIG. 7, we willnext examine the affect of changing the phase shift parameter d. FIG. 9shows a gate pulse 47 identical to the gate pulse 46 shown in FIG. 7,but shifted in the positive direction by 0.5. The parameters forgenerating the gate pulse 47 listed below are identical to those forproducing the gate pulse 46 in FIG. 7, but for the value of the phaseshift parameter d. $\begin{matrix}{d = 0.5} \\{{n(x)} = 10000} \\{{a(x)} = 16} \\{{X(x)} = x}\end{matrix}$

[0083] To produce the gate pulse 46 of FIG. 7 centered around theorigin, the phase shift parameter was set equal to 0. However, anon-zero value of phase shift parametered d causes the entire pulse tobe shifted by an amount equal to the value of the phase shift parameter.In the present example d=0.5 so the position of the pulse is shifted toleft 0.5. A phase shift like that just described may be applied to anentire series of pulses (or other window functions) when a non-zerophase shift parameter d is applied to the entire position locus functionX(x−d). This situation is shown in FIG. 10 where a sequence of pulses 49is shown. Each pulse generated corresponding to the zero crossings ofthe position locus function${{X(x)} = \frac{{Sin}\quad 2\pi \quad x}{2\pi}},$

[0084] but the actual position of each pulse is offset from thecorresponding zero crossing by 0.5.

[0085] To this point, we have explored how the window function ƒ(x)affects the output value of the DNAX output waveform within theoperational window. We have shown how the width of the operationalwindow may be varied by manipulating the operational window apertureparameter a(x). We have also examined how the position and number ofoperational windows can be established by controlling the position locusparameter X(x), and we have discussed how the position of theoperational window can be adjusted relative to the zero crossings of theposition locus function by varying the phase parameter d.

[0086] Now we will consider how the slope of the transitions at theleading and trailing edges of each pulse can be similarly altered byapplying different values or variable functions to the slew rateparameter n(x). For illustrative purposes we will examine the effects ofvarying the slew rate parameter n(x) on a DNAX function configured toproduce a single rectangular pulse. The general form of the DNAXfunction may be re-written as a rational function in the form${y(x)} = \frac{\left\lbrack {1 - \left( {{a(x)} \cdot {X\left( {x - d} \right)}} \right)^{n{(x)}}} \right\rbrack}{\left\lbrack {1 + \left( {{a(x)} \cdot {X\left( {x - d} \right)}} \right)^{n{(x)}}} \right\rbrack}$

[0087] We will use this form of the general DNAX function to illustratethe effects of changing the value of the slew rate parameter n(x). Wewish to generate a single square pulse centered on the origin having anamplitude of 2 and pulse width of 1 as well. We select the DNAXparameters as follows: $\begin{matrix}{d = 0} \\{{n(x)} = 1000} \\{{a(x)} = 2} \\{{X(x)} = x} \\{{f(x)} = 1}\end{matrix}$

[0088] The result is the square pulse 51 shown in FIG. 11. As can beseen, the pulse has an amplitude of 2 and pulse width of 1 withtransitions at ±0.5. Also, clearly visible is the fact that thetransitions are extremely steep, very nearly vertical. FIG. 11 alsoincludes three additional DNAX output waveforms 52, 54 and 56superimposed on one another along with pulse 51. The DNAX inputparameters for the additional pulses are identical, except for changesin the value of n(x). The value applied to the slew rate parameter foreach waveform is as follows: $\begin{matrix}{{{{Waveform}\quad 51}:{n(x)}} = 1000} \\{{{{Waveform}\quad 52}:{n(x)}} = 100} \\{{{{Waveform}\quad 54}:{n(x)}} = 20} \\{{{{Waveform}\quad 56}:{n(x)}} = {~~}10}\end{matrix}$

[0089] For DNAX output waveform 52, the slew rate has been diminished bya factor or 10 relative to the square wave pulse waveform 51. Thisresults in transitions of visibly shallower slope at the leading andtrailing edges of the pulse. Reducing the slew rate by an additionalfactor of 5 results in the considerably rounded pulse of DNAX outputwaveform 54. And a slew rate of just 10 results in the very gentle riseand fall of the pulse of DNAX output waveform 56.

[0090] The last input variable to consider for the general form of theDNAX function is the weight factor b(x). The weight factor b(x) is afactor which determines the linearity of the function ƒ(x) producedwithin the operational window. Until now, the weight factor b(x) hasbeen set equal to the constant value 1. By changing the weight factorb(x) one may alter the function ƒ(x) within the operational windows inpredictable ways. When the weight factor b(x)=1, the function ƒ(x) isreproduced within the operational window unchanged. With other values ofb(x), however, the output function ƒ(x) is distorted.

[0091]FIGS. 12, 13 and 14 illustrate the consequences of changing theweight factor on a simple function ƒ(x)=x. The output waveform 58 shownin FIG. 12 was generated with DNAX parameters: d = 0.5 n(x) = 1000a(x) = 2 X(x) = x f(x) = x b(x) = 1

[0092] note that the center of the operational window is shifted to theright by 0.5 due to the phase shift parameter d. The width of theoperational window remains 1 due to the operational window apertureparameter value of 2 and the position locus parameter value X(x)=x. InFIG. 12, the weight factor is set to b(x)=1. Since b(x)=1 there is nodistortion of the output waveform within the operational window 60. Theoutput waveform 60 within the operational window is the straight rampfunction ƒ(x)=x. In FIG. 13 the weight factor has been adjusted tob(x)=0.5, resulting in the output waveform 62. The 0.5 weight factorresults in output values greater than ƒ(x) within the operational window64. Similarly, in FIG. 14 the weight factor has been adjusted to b(x)=2.The result is the output waveform 66. The 2.0 weight factor results inoutput values less than ƒ(x) within the operational window 68. Thedistortion added to the window function ƒ(x) is solely a function of theweight factor b(x). By measuring the amount of distortion in a receivedsignal, it is possible to determine the applied weight factor.Furthermore, as with the other DNAX input parameters, the weight factorb(x) may be set equal to a constant or may be set to a variablefunction. Thus, it is possible to add additional information to a datasignal by altering the distortion added by a modulating function appliedto the weight factor b(x).

[0093] We now turn to a number of applications where the DNAX functionmay be applied to generate useful waveforms and modulated signals. Thepresent invention involves applying the DNAX function in a number ofdifferent ways to generate different waveforms (i.e. signals) on whichdata may be encoded, or whose shape has some other significance. In somecases the DNAX function may be employed to implement known modulationtechniques in an easier more efficient manner. In other cases the DNAXfunction may be employed to create entirely new modulation and data andcoding techniques. Furthermore, the versatility of the DNAX techniqueallows multiple coding and modulation techniques to be applied to asingle waveform, thereby greatly increasing the density of data that canbe carried by a single signal.

[0094] According to an embodiment of the invention, the DNAX functionmay be employed to generate a pulse width modulated (PWM) waveform. PWMis a well know modulation technique in which a series of regularlyspaced rectangular pulses is modulated by varying the width of eachpulse according to the magnitude of a reference or modulating signal.

[0095]FIG. 15 shows a pulse width modulated waveform 100 which has beenmodulated by a sinusoidal modulating waveform 102 shown in FIG. 16. Ascan be seen, the width of the individual pulses such as pulses 104, 106,108 and 110 of the waveform 100 vary according to the correspondingmagnitude of the sinusoidal modulating signal 102. Pulses correspondingto the peak of the sinusoidal modulating signal (such as pulse 104) havea relatively short duration, while those pulses corresponding to theminimum values of the signal 102 (such as pulse 110) are considerablywider. The width of the pulses corresponding to intermediate values ofthe modulating signal 102 such as pulses 106, 108 have an intermediateduration directly related to the corresponding magnitude of themodulating signal. In this example, pulse width has an inverserelationship with the magnitude of the modulating signal 102. The pulsewidth decreases with the increasing magnitude of the modulation andincreases with decreasing magnitude. Of course, waveform 100 could havebeen modulated in exactly the opposite manner, with wider pulsescorresponding to higher values of the modulating signal, and narrowerpulses corresponding to lower values.

[0096] In any case, the PWM waveform 100 was generated according to thepresent invention using the DNAX function described above. In generatinga PWM waveform using the DNAX function, it is first necessary todetermine the characteristics of the modulating signal and the desiredcharacteristics of the PWM output waveform. For the example shown inFIG. 15, the modulating waveform, or signal, is a sinusoid having afrequency of 10 khz. The modulating signal is offset, having a peakvalue of 4, a minimum of 0 and an average value of 2. Thus, themodulating signal s(t) may be represented by the equation

s(t)=2 Sin(2π·10·10³ ·t)+2

[0097] Since the frequency of the modulating signal is expressed in khzor thousands of cycles per second, we have replaced the independentvariable x with t indicating that the abscissa axis in the accompanyingplots relate to time.

[0098] Next, we define the characteristics of the PWM output signal andthe relationship between the individual pulses of the PWM outputwaveform and the modulating signal s(t). According to the example shown,the PWM output waveform is formed of a plurality of rectangular pulseseach having a peak amplitude of 1 and a minimum value of 0. Therectangular pulses occur periodically at a frequency of 80 khz. We startby generating a sequence of unmodulated pulses having the desiredcharacteristics, i.e., rectangular pulses having an amplitude of 1 andoccurring periodically at a frequency of 80 khz as shown at 116 in FIG.17. The output waveform 116 can be generated using the general form ofthe DNAX functiony(t) = [a(t)X(x − d)^(n(t)) + f(t)]^([b(t) − (a(t)X(x − d))^(n(t))])

[0099] Since we desire pulses having an amplitude of 1 with notdistortion, we set ƒ(t)=1 and b(t)=1. Further, since we desire steeptransitions at the leading and trailing edges of each pulse, we set n(t)equal to some arbitrarily large constant such as 1000. The pulses are tobe periodic so we must set the position locus parameter X(x−d) equal toa periodic function having zero crossings which occur at the desiredpulse frequency. Using the DNAX function to generate pulses, the pulsesoccur at each zero crossing of the position locus parameter. A sinusoidhas two zero crossings per period. Therefore, we may select a sinewavehaving a frequency equal to one half the frequency of the desired pulsefrequency as the position locus function. For convenience, we selectX(t−d)=⅓ Sin ωt where ω=2πƒ and ƒ=40 khz. Also since there is no need toshift the location of the pulses relative to the zero crossing of theposition locus parameter we may set the phase shift parameter d=0. Theposition locus function ⅓ Sin ωt is shown at 114 in FIG. 17. Forclarity, position locus function 114 is scaled by a factor of 10 so asto not visually interfere with pulse sequence 116. With these values thegeneral form of the DNAX function becomes${y(t)} = \left\lbrack {\left( {{a(t)}\frac{{Sin}\left( {\omega \quad t} \right)}{3}} \right)^{10^{3}} + 1} \right\rbrack^{\lbrack{1 - {{a{(t)}}{\lbrack\frac{{Sin}{({\omega \quad t})}}{3}\rbrack}}^{10^{3}}}\rbrack}$

[0100] To generate the pulse sequence 116 shown in FIG. 17 where eachpulse is the same width, a(t) is set equal to a constant rather than atime variable modulating function. However, to convert the unmodulatedwaveform 116 of FIG. 17 into the PWM modulated waveform 100 of FIG. 15it is necessary to modulate the operational window aperture parametera(t) according to the modulation signal s(t) 102.

[0101] Central to the concept of pulse width modulation is determiningthe width of each pulse based on a corresponding value of the modulatingsignal s(t). Related to the pulse width is concept of duty cycle. Theduty cycle of a pulse may be expressed as the percentage of time thatthe pulse is “on” (equal to 1) during the course of an entire pulseperiod. Thus, a pulse having a 50% duty cycle will be on for one half ofa pulse period and off for the other half. Clearly, altering the dutycycle will have a corresponding effect on the pulse width.

[0102] In the present example, we wish to modulate the duty cycle w(t)of each pulse according to the modulating signal s(t) as follows. Whenthe modulating signal is at its peak value, s(t)=4, we want the outputpulses to have a 5% duty cycle. When the modulating signal is at itsminimum value, s(t)=0, we want the pulses to have an 85% duty cycle.Furthermore, we want the duty cycle to vary linearly between 5% and 85%for intermediate values of the modulating signal, i.e., 0<s(t)<4. Thisrelationship may be expressed in the linear transfer function

w(t)=−0.2s(t)+0.85

[0103] which is shown graphically at 112 in FIG. 18.

[0104] The duration of each pulse is related to the duty cycle w(t) inthat the pulse duration is equal to the pulse period multiplied by thevalue of the desired duty cycle. Using DNAX, the frequency of theposition locus parameter determines the frequency of, and therefore theperiod, of each pulse. In this case the frequency of the position locusparameter is ƒ=40×10³ hz. Since the period the period T of the positionlocus function is equal to the reciprocal of the frequency$T = \frac{1}{f}$

[0105] and since there are two pulses for each cycle of the positionlocus function, the pulse period is equal to one half the period of theposition locus function ${Tp} = {\frac{T}{2}.}$

[0106] Thus, ${{Pd}(t)} = {{w(t)}{\frac{T}{2}.}}$

[0107] Using the DNAX function to generate a series of pulses, the widthof the operational window defines the width of each pulse. Bydefinition, the width of the operational window (or windows as the casemay be) created by the DNAX function is determined by the equation${{Pd}(t)} = {\frac{2}{{a(t)}{\frac{{X(t)}}{t}}}.}$

[0108] Thus, the width of each pulse is inversely proportional to thevalue of the operational window aperture parameter a(t) and the absolutevalue of the derivative of the position locus function X(x) at the zerocrossings of the position locus function. By altering the value of a(t)in accordance with duty cycle transfer function w(t) we can vary thewidth of the output pulses based on the modulating signal s(t).

[0109] We have already defined the pulse position locus parameter as${X(t)} = {\frac{1}{3}{Sin}\quad \omega \quad t}$

[0110] and the phase shift parameter as

d=0.

[0111] Thus,${\frac{{X\left( {t - d} \right)}}{t}} = {{\frac{{\frac{1}{3}}{Sin}\quad \omega \quad t}{t}} = {{\frac{\omega}{3}\cos \quad \omega \quad t}}}$

[0112] The relationship between the position locus parameter${X(t)} = {\frac{1}{3}{Sin}\quad \omega \quad t}$

[0113] Sin ωt and the absolute value of its derivative with respect totime, ${\frac{{X(t)}}{t}},$

[0114] is shown in FIG. 19 where the output waveform 114 shows${X(t)} = {\frac{1}{3}{Sin}\quad \omega \quad t}$

[0115] and the output waveform 115 shows ${\frac{{X(t)}}{t}}.$

[0116] As can be seen, at the points where X(t) crosses 0(P_(i)(x_(i):0)) |cos ωt|=1, and${{\frac{}{t}{X(t)}}} = {\frac{\omega}{3}.}$

[0117] Thus, the equation for determining the pulse duration Pd becomes${{Pd}(t)} = {{\frac{2}{{a(t)}\frac{\omega}{3}}\quad {or}\quad {{Pd}(t)}} = {\frac{6}{{a(t)} \cdot \omega}.}}$

[0118] Since we have already determined that the pulse duration Pd(t) isequal to the duty cycle function w(t) multiplied by one half the periodof the position locus function$\frac{T}{2},\quad {{{Pd}(t)} = {{w(t)}\frac{T}{2}}}$

[0119] and single the duty cycle function w(t) may be determined by thelinear transfer function

w(t)=−0.2s(t)+0.85,

[0120] we may write${{Pd}(t)} = {{\left( {{{- 0.2}{s(t)}} + 0.85} \right)\frac{T}{2}} = {\frac{6}{{a(t)}\omega}.}}$

[0121] Solving for a(t) we obtain${a(t)} = {{\frac{6}{\left( {{{- 0.2}{s(t)}} + 0.85} \right)\frac{T}{2}\omega} \cdot \frac{1}{2f}}2\pi \quad {f.}}$

[0122] Since ${T = \frac{1}{f}},$

[0123] and ω=2πƒ, the above equation becomes${a(t)} = \frac{6}{\left( {{{- 0.2}{s(t)}} + 0.85} \right)\pi}$

[0124] Using standard polynomial expansion techniques the above equationmay be expressed as:

a(t)=2.247+0.2s(t)²+0.14s(t)³+14·10⁻¹⁰ s(t)¹⁷

[0125] This function is shown in FIG. 20. As can be seen, the outputfunction has a peak value of 38.197 and a minimum value of 2.247. Insome cases, as in the present case, it may be desirable to maintain thevalue of a(t) above some minimum threshold value to prevent overmodulation and over-lapping pulses. For example, to ensure adequatespacing between pulses, a(t) may be limited to values greater than 3,a(t)>3. In the present example this may be accomplished by adding anarbitrary constant to the equation for determining a(t). The constant isof sufficient magnitude to raise the minimum value of a(t), above theminimum threshold a(t)>3. In the present example, we select a constant$c = {\frac{\pi}{4}.}$

[0126] Thus, the final equation for determining the modulated value ofa(t) becomes${a(t)} = {2.247 + {0.2{s(t)}^{2}} + {0.14{s(t)}^{3}} + {{14 \cdot 10^{- 10}}{s(t)}^{17}} + {\frac{\pi}{4}.}}$

[0127] This function may be referred to as the modulation indexfunction. It is plotted along with the sinusoidal modulation signal s(t)102 in FIG. 21. Since the pulse duration is inversely related to thevalue of a(t), the duty cycle of the output pulses is narrower forhigher values of a(t). Comparing the modulation index function a(t) 120to the modulation signal s(t) 102, it can be seen that a(t) reaches itsgreatest value coincidentally with the peaks of the modulation signals(t), and its lowest value at the minimum values of the sinusoidalmodulation signal s(t). Substituting the above modulation index functiona(t) into the DNAX function,${y(t)} = \left\lbrack {{{{a(t)} \cdot \frac{1}{3}}{{Sin}\left( {2{\pi \cdot 40 \cdot 10^{3}}t} \right)}^{100}} + 1} \right\rbrack^{\lbrack{1 - {\lbrack{{a{(t)}}\frac{1}{3}{{Sin}{({2{\pi \cdot 40 \cdot 10^{3}}t})}}}\rbrack}^{100}}\rbrack}$

[0128] produces the PWM waveform 100 shown in FIG. 15.

[0129] Pulse width modulation is but one of the myriad modulationtechniques that may be implemented using the DNAX technique. By astutemanipulation of the parameters d, n(x), a(b), X(x), ƒ(x), and b(x),communication, data, power and other signals may be modulated andcomplex waveforms generated in a very straight forward manner.Additionally, modulation techniques may be combined to greatly increasethe amount of data that may be encoded onto a single signal. The outputwaveforms of DNAX functions may be input to parameters of other DNAXfunctions, acting as modulating signals for other DNAX functions. Suchnesting or serial relationships further enhance the usefulness andcomplexity of the output waveforms that may be generated using the DNAXfunction. In addition to known modulation schemes, the power of the DNAXfunction opens the door to new modulation schemes which heretofore wouldnot have been practicable.

[0130] Slew rate modulation, is a new modulation technique made possibleby the DNAX function. Recall that according to the DNAX technique, slewrate (represented by the parameter n(x) defines the speed with whichtransitions occur in the output waveform between those portions of theoutput waveform lying outside an operational window and those within.Employing the DNAX technique to generate a series of pulses, amodulating signal may be input to the slew rate parameter n(x) so thatmeaningful data may be encoded on the leading or trailing edges of eachpulse.

[0131] For purposes of illustration we will describe an example of slewrate modulation (SRM) wherein a 50% duty cycle rectangular wave signalwill be modulated onto the leading and trailing edges of the pulses of asecond rectangular carrier signal. The carrier signal will have anamplitude of 1, a frequency of 2 khz, and 50% duty cycle. It should beunderstood, however, that SRM may be applied to carrier signals otherthan a 2 khz, 50% duty cycle rectangular wave carrier, and that signalsother than the 2 khz 50% duty cycle rectangular wave modulating signalmay be encoded using this technique.

[0132] As with PWM, or any other modulation scheme employing the DNAXfunction, we begin with the general form of the DNAX function:y(t) = [[a(t) ⋅ X(t − d)]^(n(t)) + f(t)]^([b(t) − a(t) ⋅ X(t − d)^(n(t))])

[0133] As has already been described, the parameter n(t) determines theslew rate. Thus, by applying a modulating signal to the input parametern(t), the slew rate of the output waveform y(t)will vary according tothe modulating signal.

[0134] In the present example the modulating signal is a rectangularwave signal which itself must be generated using to the DNAX function.Thus, we set the slew rate parameter n(t) of our DNAX output equationequal to a first DNAX function adapted to produce the desired modulatingrectangular wave output waveform:n(t) = Q[[a(t)X(t − d)]^(m(t)) + f(t)]^([1 − (a(t)X(t − d))^(m(t))]) + C

[0135] The DNAX parameters for the modulating waveform n(x) are set asfollows: $d = \frac{\pi}{4}$ X(t) = Sin(2π × 10³t)m(t) = 100  Q = 500 a(t) = 1.5  C = 5 f(t) = 1 b(t) = 1

[0136] The result of this DNAX function is the rectangular pulse outputsignal 128 shown in FIG. 22. The position locus signal 20 Sin 2πx10³t124 is plotted along with square wave output n(t). As with previousexamples, a rectangular pulse 122 is generated for each zero crossing ofthe position locus function X(t)=Sin 2πx10³t 124. The periodic pulsetrain has a frequency of 2 khz and a 50% duty cycle. Furthermore, eachpulse has a maximum value of 505 and a minimum value of 5 due to themultiplier Q and the offset constant C.

[0137] As we have noted, the square wave signal n(t) 128 showngraphically in FIG. 22 represents the modulating signal to be applied tothe slew rate parameter of our carrier waveform. The carrier signal hasthe same frequency as the modulating signal. Thus, the carrier signal isa 2 khz square signal wave signal with an amplitude varying between 0and 1 and a 50% duty cycle. Again we start with the general form of theDNAX functiony(t) = [(a₁(t)X(t − d))^(n(t)) + f(t)]^([1 − a₁(t)X(t − d)^(n(t))]).

[0138] The DNAX parameters for the SRM modulated carrier signal are setas follows: d = 0n(t) = Q[a(t)X(t − d)^(m(t)) + 1]^([1 − a(t)X(t − d)^(m(t))]) + Ca(t) = 2 X(t) = Sin  ω  t; ω = 2π  f; f = 2 × 10³ f(t) = 1b(t) = 1

[0139] The output waveform 126 generated by the DNAX function with theabove set of input parameters is shown in FIG. 23. The modulating signaln(t) 128 (divided by 10 to fit the scale shown) is also plotted forreference, as is the position locus function Sin 2π·10³t 124. As canbest be seen graphically, the leading edge 130 of each pulse of theoutput waveform 126 falls in an area 134 which coincides with a peak inthe modulating signal n(t) 128. In contrast, the trailing edges 132 ofeach pulse coincide with an area 136 where n(t) is at its minimum value.It is also clear that where the modulating signal n(t) is at its peakthe overall output function y(t) 126 has the relatively high slew rate,505, and where n(t) is at its minimum value, the output function y(t)126 has the relatively low slew rate of 5. Because the leading edges 130of the output pulses coincide with areas of high slew rate 134, theleading edge transitions 130 are extremely fast, leading to very steepedges. On the other hand, because the trailing edges of the pulses 132coincide with areas of low slew rate, the trailing edge transitions 132occur more slowing, resulting in the gentler slope of the trailing edges132. This effect is clearly illustrated in FIG. 23. The leading edges ofeach pulse appear substantially vertical, and the trailing edges appearwith a pronounced slope.

[0140] By altering the value of the phase shift component d in themodulating signal n(t), the pulses of the modulating signal can be phaseshifted horizontally, thereby significantly changing the modulatedcharacteristics of the output waveform y(t). For example, FIG. 24 showsan SRM waveform 138 in which the phase shift component of the modulatingsignal has been removed. The same DNAX output waveform 138 was generatedwith the same equations as in the previous example, but with d=0 appliedto the phase shift parameter of the modulating signal n(t). As can beseen the rectangular pulses of the modulating signal n(t) 128 haveshifted so that they now coincide with the zero crossing of the positionlocus signal 124. Now both the leading and trailing edges 130, 132 ofthe output pulses of the output waveform y(t) occur in areascorresponding to the peak value in the modulating signal 134. Thus, bothleading and lagging transitions occur at the higher slew ratecorresponding to faster transitions.

[0141] Similarly, the output waveform 140 shown in FIG. 25 shows themodulating signal n(t) 128 phase shifted $\frac{\pi}{6}$

[0142] rather then $\frac{\pi}{4}$

[0143] or 0. In this example, the trailing edges 132 of the outputpulses fall in the area corresponding to the peak value of slew ratesignal 134 and the leading edges are located in the troughs 136. Thus,in this example the leading edge transitions 130 take place at theslower slew rate n(t)=5, and the trailing edges 132 occur at the fasterslew rate 505.

[0144] Finally, FIG. 26 shows a similar SRM modulated waveform 142, butwhere the duty cycle of the modulation signal n(t) 128 has been reducedby increasing the value of a(t) in the DNAX function generating themodulation signal. The result is that both the leading and trailingedges 130, 132 of the output pulses of the output waveform y(t) 142occur at locations coinciding with the minimum value of the slew rateparameter n(t). Thus, both the leading and lagging transitions occur atthe slower slew rate.

[0145] By varying the slew rate parameter in different ways any numberof pulse shapes may be obtained. A sampling of various pulse shapesobtained using the SRM modulating technique just described is shown inFIG. 27.

[0146] The SRM modulating technique may be implemented in functionalblock form as shown in FIG. 28. The functional block diagram 144 of FIG.25 includes two DNAX function blocks or “cells” 146, 148. Each cellrepresents an implementation of a DNAX function. The first cell 144represents the DNAX function for generating the modulating signal n(t).The inputs to the first cell 144 include the phase shift parameter d150, the slew rate parameter n₁(t) 152, the pulse width parameter a₁(t)154 and the pulse position locus function X(t) 160. The window functionƒ(t) (not shown) is set equal to 1. The output of the first cell is then₂(x) parameter 158 input into the second cell 146. Also, input to thesecond cell 148 is a phase shift parameter d 160, a second pulse widthparameter a₂(x) 162 and the pulse position locus function X(t) 152. Theoutput of the second cell 144 represents the desired SRM waveform 164.

[0147] In this specification we have discussed in depth two separatemodulating techniques employing the DNAX function. From these examples,it should be clear that many additional modulation techniques may beadvantageously implemented via the DNAX function. Appropriatemanipulation of the DNAX function's input parameters can result in datamodulation in any number of different ways. For example, in addition tothe PWM and SRM modulation techniques described above, the DNAX functionmay also be employed to implement pulse position modulation, timedivision multiplexing (TDM), code division multiple access (CDMA), highdensity high volume data transmission, polynomial encryption, and othersignal formats.

[0148] It should be understood that various changes and modifications tothe presently preferred embodiments described herein will be apparent tothose skilled in the art. Such changes and modifications can be madewithout departing from the spirit and scope of the present invention andwithout diminishing its intended advantages. It is therefore intendedthat such changes and modifications be covered by the appended claims.

The invention is claimed as follows:
 1. A method of generating awaveform having a defined output value for all input values, thewaveform including at least one operational input window wherein theoutput value of said waveform is substantially equal to a base value forsubstantially all input values outside said at least one operationalwindow and the output value of the waveform is determined by a windowfunction for substantially all input values within said at least oneoperational window, the method comprising the steps of; providing acontinuous function that includes said operational window function, aposition locus parameter for determining a position of said operationalwindow, an operational window aperture parameter for determining a widthof said operational window, and a slew rate parameter for determiningthe slope of transition rate for output values of said function fromsaid base value to a value based on said operational window function forinput values near at least one edge of said operational window; andselecting said operational window function and said position locusparameter, said operational window aperture parameter and said slew rateparameter to achieve a desired waveform.
 2. The method of generating awaveform of claim 19, wherein said function is of the formy(x) = [(a(x)X(x − d))^(n(x)) + f(x)]^([b(x) − (a(x) ⋅ X(x − d))^(n(x))])

wherein ƒ(x) is said window function, X(x−d) is said position locusparameter, a(x) is said operational window parameter, and n(x) is saidslew rate parameter.
 3. The method of generating a waveform of claim 2further comprising the step of setting the window function ƒ(x) equal toa constant such that said waveform includes a pulse having a pulseamplitude equal to said constant.
 4. The method of generating a waveformof claim 3 further comprising the step of setting the position locusparameter X(x−d) equal to a function having a plurality of abscissaintercepts, such that said waveform includes a plurality of said pulsescorresponding to said x-intercepts.
 5. The method of generating awaveform of claim 4 wherein said function having a plurality of abscissaintercepts is a periodic function.
 6. The method of generating awaveform of claim 4 further comprising the step of setting theoperational window aperture parameter equal to a variable function suchthat the width of said pulses varies according to said variablefunction.
 7. The method of generating a waveform of claim 4 furthercomprising the step of setting said slew rate parameter n(x) equal to avariable function such that the slope of the transitions between saidbase value and said pulse amplitude varies according to the value ofsaid variable function.
 8. The method of generating a waveform of claim1, wherein said function is of the formy(x) = [(a(x)X(x − d))^(n(x)) + 1]^([−(a(x)X(x − d))^(n(x))])


9. The method of generating a waveform of claim 1, wherein said functionis of the form${y(x)} = \frac{\left\lbrack {1 - \left( {{a(x)} \cdot {X\left( {x - d} \right)}} \right)^{n{(x)}}} \right\rbrack}{\left\lbrack {1 + \left( {{a(x)} \cdot {X\left( {x - d} \right)}} \right)^{n{(x)}}} \right\rbrack}$


10. A method of generating a modulated waveform comprising the steps of:providing a continuous function for generating an output waveform, saidfunction defining at least one operational window where an output valueof said function is substantially equal to a base value forsubstantially all input values outside said operational window, saidfunction defined by: a first parameter for establishing a position ofsaid operational window; a second parameter for establishing a width ofsaid operational window; a third parameter for establishing an outputvalue of said function for substantially all input values within saidoperational window; and a fourth parameter for establishing the slope ofthe output waveform during transitions between output values of saidfunction corresponding to input values outside the operational windowand output values of said function corresponding to input values withinthe operational window; and applying a modulating signal to at least oneof said first, second, third and fourth parameters such that the valueof the parameter to which the modulation signal is applied variesaccording to the modulation signal.
 11. The method of generating amodulated waveform according to claim 10, further comprising the stepof: applying a periodic function to said first parameter such that aplurality of operational windows is created corresponding to the periodof said first periodic function.
 12. The method of generating amodulated waveform according to claim 11, wherein the first periodicfunction is an angle modulated signal such that the position of theoperational windows varies according to changes in at least one of thephase and frequency of the first periodic input function.
 13. The methodof generating a modulated waveform according to claim 11, wherein themodulating signal is applied as an input to said second parameter suchthat the width of said operational windows varies according to themodulation signal.
 14. The method of generating a modulated waveformaccording to claim 11, wherein the modulating signal is applied as aninput to said third parameter, such that the output value of saidfunction within the operational window is itself a modulated signal. 15.The method of generating a modulated waveform according to claim 12,wherein said modulating signal is an amplitude modulated signal.
 16. Themethod of generating a modulated waveform according to claim 14, whereinsaid modulating signal is an angle modulated signal.
 17. The method ofgenerating a modulated waveform according to claim 14, wherein saidmodulation signal is a digitally coded signal.
 18. The method ofgenerating a modulated waveform according to claim 11, wherein themodulation signal is applied as an input to said fourth parameter suchthat the slope of the output waveform during transitions between outputvalues of said function corresponding to input values outside theoperational windows and output values of said function corresponding toinput values within the operational windows varies according to themodulation signal.
 19. The method of generating a modulated waveformaccording to claim 10, wherein said function is of the form:y(x) = [(a(x) ⋅ X(x − d))^(n(x)) + f(x)]^([b(x) − (a(x)X(x − d))^(n(x))])

where: a(x) is the first parameter X(x−d) is the second parameter ƒ(x)is the third parameter; and n(x) is the fourth parameter.
 20. The methodof generating a modulated waveform of claim 19 where b(x) is a weightingfactor affecting the linearity of the operational window function withinthe operational window and wherein a modulating signal is applied as aninput to b(x).
 21. The method of generating a modulated waveform ofclaim 19 where d is a phase shift factor for adjusting the position ofthe operational window relative to the position determined by the firstparameter.
 22. A method of generating one or more operational windows,said one or operational windows having a window value and at least onetransition between the window value and a base value, the methodcomprising the steps of: providing a first function having definedoutput values for all input values within a continuous input domain,said function including: a position locus parameter for defining alocation of said one or more operational windows; an aperture parameterfor determining a pulse width of said one or more operational windows; awindow function for determining the window value; a slew rate parameterfor determining a slope of said at least one transition; and providing asecond function having at least one input intercept; and applying theoutput value of the second function as an input to said first positionparameter, said first function being adapted to generate an operationalwindow based on each input intercept of said second function.
 23. Themethod of claim 22 wherein said function is of the form:y(x) = [(a(x) ⋅ X(x − d))^(n(x)) + f(x)]^([b(x) − (a(x) ⋅ X(x − d))^(n(x))])

where: d corresponds to a phase shift of the one or more pulses relativeto one or more positions determined by the position locus parameter;n(x) corresponds to said slew rate parameter; a(x) corresponds to saidaperture parameter; X(x) corresponds to said position locus parameter;ƒ(x) corresponds to said window function; and b(x) corresponds to aweighting factor affecting the linerity of the window function.
 24. Themethod of claim 23 wherein X(x) comprises a function having a pluralityof x intercepts and ƒ(x) equals a constant such that a plurality of flatpulses are generated corresponding to the x-intercepts of X(x), eachpulse having an amplitude equal to the constant.
 25. The method of claim24 wherein X(x) is a frequency modulated function.
 26. The method ofclaim 24 wherein X(x) is a phase modulated function.
 27. The method ofclaim 24 wherein an output of a variable function is input to saidaperture parameter a(x) such that said plurality of pulses is pulsewidth modulated according to said variable function.
 28. The method ofclaim 24 wherein an output of a variable function is input to said slewrate parameter n(x) such that the transitions between the base value andthe ƒ(x) constant value are slope rate modulated.
 29. A method ofcreating a pulse width modulated waveform comprising the steps of:providing a DNAX function; applying a periodic function as an input to aposition locus parameter of the DNAX function to thereby generating aplurality of pulses; providing a modulating signal; calculating amodulation index function for establishing a relationship between themagnitude of the modulating signal and a corresponding modulated outputpulse width; and applying the modulation index function to a pulse widthparameter of the DNAX function to modulate the width of said pluralityof pulses.
 30. A method of generating a slew rate modulated waveformcomprising steps of: providing a first function of the formn(x) = Q  [(a₁X(x − d))^(M) + 1]^([1 − (a ⋅ X(x − d))^(M)])

selecting values for Q, a, X(x), d and M to generate a desiredmodulating waveform; providing a second function of the formy(x)=  [(a₂X(x − d))^(n(x)) + 1]^([1 − (a₂X(x − d))^(n(x))])

where n(x) is the output waveform from said first function.